用于指定输入波前和目标辐照度的单个自由曲面设计
Single freeform surface design for prescribed input wavefront and target irradiance
摘要：
在光束整形应用中，用于光束整形过程的必要光学元件的数量的最小化可以有益于光学系统的紧凑性并降低其成本。因此，对于既不是平面也不是球面的输入波前的单个自由曲面设计是感兴趣的。在这项工作中，设计一个给定的零展度源和复杂的目标辐照度的单个自由曲面。因此，不仅假设准直输入光束或点源。相反，考虑源平面上的预定义输入射线方向矢量场和辐照度分布，其必须由单个自由曲面重新分布以给出预定义目标辐照度。为了解决这个设计问题，偏微分方程（PDE）或PDE系统，分别为未知的表面和其相应的光线映射来自能量守恒和光线跟踪方程。与以往单一自由曲面设计问题的偏微分方程不同，本文推导的Monge-Ampère型偏微分方程是在直角坐标系下针对一般零扩展源的。偏微分方程系统离散与有限差分和由此产生的非线性方程组求解的求根算法。PDE系统的有效解决方案的基础上建立了一个初始的输入方向矢量场，它使用最佳质量运输与二次成本函数的引入constituent方法。在详细描述了数值算法之后，通过将其应用于几个设计实例，证明了设计方法的有效性。这包括超出近轴近似的准直输入光束的重新分布、点源辐射的成形以及将准直输入波前成形为复杂的目标辐照度分布。
英文摘要：
In beam shaping applications, the minimization of the number of necessary optical elements for the beam shaping process can benefit the compactness of the optical system and reduce its cost. The single freeform surface design for input wavefronts, which are neither planar nor spherical, is therefore of interest. In this work, the design of single freeform surfaces for a given zero-étendue source and complex target irradiances is investigated. Hence, not only collimated input beams or point sources are assumed. Instead, a predefined input ray direction vector field and irradiance distribution on a source plane, which has to be redistributed by a single freeform surface to give the predefined target irradiance, is considered. To solve this design problem, a partial differential equation (PDE) or PDE system, respectively, for the unknown surface and its corresponding ray mapping is derived from energy conservation and the ray-tracing equations. In contrast to former PDE formulations of the single freeform design problem, the derived PDE of Monge-Ampère type is formulated for general zero-étendue sources in cartesian coordinates. The PDE system is discretized with finite differences and the resulting nonlinear equation system solved by a root-finding algorithm. The basis of the efficient solution of the PDE system builds the introduction of an initial iterate constuction approach for a given input direction vector field, which uses optimal mass transport with a quadratic cost function. After a detailed description of the numerical algorithm, the efficiency of the design method is demonstrated by applying it to several design examples. This includes the redistribution of a collimated input beam beyond the paraxial approximation, the shaping of point source radiation and the shaping of an astigmatic input wavefront into a complex target irradiance distribution.
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